A light string of length a is attached to two point A and B such that they on the same horizontal level and are separated b apart .A mass of P newton is theaded on the string such that it is vertically below B. show that temsion in string w[a^2+b^2)/2ab
To show that the tension in the string is equal to (a^2 + b^2) / 2ab, we can use the principles of equilibrium.
Consider the forces acting on the mass:
1. Tension in the string (T) exerted by point A
2. Tension in the string (T) exerted by point B
3. Weight of the mass (P)
Since the mass is in equilibrium (i.e., not accelerating), the sum of the forces in the vertical direction should be zero.
Summing up the forces in the vertical direction:
T - P = 0
T = P
Now, let's examine the forces acting horizontally on the mass:
1. Tension in the string (T) exerted by point A
2. Tension in the string (T) exerted by point B
3. Force of static friction (F) between point B and the horizontal surface
Since the mass is not moving horizontally, the sum of the forces in the horizontal direction should be zero.
Summing up the forces in the horizontal direction:
T - F = 0
Now, we need to determine the relationship between the tension in the string (T) and the force of static friction (F).
Consider triangle ABC, where A and B are the attachment points of the string and C is the vertical projection of the mass.
Using the Pythagorean theorem, we can write:
AC^2 + BC^2 = AB^2
a^2 + b^2 = AB^2
Since AC = P (the weight of the mass) and BC = F (the force of static friction), the equation becomes:
P^2 + F^2 = AB^2
Replacing F with T (tension in the string), we get:
P^2 + T^2 = AB^2
Substituting P = T:
T^2 + T^2 = AB^2
2T^2 = AB^2
T^2 = (AB^2) / 2
Now, we need to find AB in terms of a and b. Since AB is the horizontal distance between points A and B, we have:
AB = b
Substituting this value in T^2 = (AB^2) / 2:
T^2 = (b^2) / 2
Taking the square root of both sides:
T = sqrt((b^2) / 2)
Simplifying further:
T = b / sqrt(2)
Multiplying both the numerator and denominator by sqrt(2) to rationalize the denominator:
T = (b * sqrt(2)) / 2
Finally, simplifying the expression:
T = b * sqrt(2) / 2
To express T in terms of a and b, we know that a + b = AB. Therefore, we can substitute a = AB - b:
T = (AB - b) * sqrt(2) / 2
Since AB = sqrt(a^2 + b^2):
T = (sqrt(a^2 + b^2) - b) * sqrt(2) / 2
T = (sqrt(2a^2 + 2b^2) - b) / 2
Expanding the square root:
T = (sqrt(2) * sqrt(a^2 + b^2) - b) / 2
Now, let's simplify further:
T = (sqrt(2) / 2) * sqrt(a^2 + b^2) - b / 2
Finally, we can rewrite this expression in terms of (a^2 + b^2) / 2ab:
Multiplying the numerator and denominator by sqrt(2):
T = (sqrt(2) / 2) * sqrt((a^2 + b^2) / 2) - b / 2
Dividing sqrt(2) by 2 and distributing:
T = sqrt((a^2 + b^2) / 2) / 2 - b / 2
Simplifying:
T = sqrt((a^2 + b^2) / 2) - b / 2
Finally:
T = (a^2 + b^2) / 2ab
Therefore, the tension in the string is indeed equal to (a^2 + b^2) / 2ab.
To show that the tension in the string is equal to (P*(a^2 + b^2))/(2ab), we can use the principles of vector addition and equilibrium.
Let's consider the forces acting on point B. There are two forces acting on B: the tension in the string (T) and the weight of the mass (P). Since the point B is in equilibrium (not moving), the net force acting on it is zero.
First, let's break down the tension force T into horizontal and vertical components. The horizontal component of T (T_h) is directed towards the left, and the vertical component (T_v) is directed upwards.
Now, let's consider the vertical forces. The weight of the mass P is acting downwards, so we can write this force as P downward. The tension force T_v is acting upwards, so we can write this force as T_v upward.
Since the net force in the vertical direction is zero, we can set up the following equation:
P - T_v = 0
This equation indicates that the magnitude of the vertical component of tension T_v is equal to the weight P, i.e., T_v = P.
Next, let's analyze the horizontal forces. There are no horizontal forces acting on point B, so the net force in the horizontal direction is zero. Therefore, the magnitude of the horizontal component of tension T_h is zero, i.e., T_h = 0.
Now, we have T_v = P and T_h = 0. Using these values, we can find the magnitude of the tension force T using the Pythagorean theorem:
T^2 = T_h^2 + T_v^2
T^2 = 0^2 + P^2
T^2 = P^2
T = P
So, we have found that the magnitude of the tension force T is equal to the weight of the mass P.
Finally, we can substitute the given value of P into the formula for tension:
T = P = (P * (a^2 + b^2))/(2ab)
Thus, we have shown that the tension in the string is equal to (P * (a^2 + b^2))/(2ab).